334 Control volume integral analysis of the continuity equation We will now

334 control volume integral analysis of the

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3.3.4 Control volume (integral) analysis of the continuity equation We will now introduce a much simpler approach than attempting to solve the PDE that is the continuity equation. We will see that this can often be carried out analytically and, although it does not always produce exact results, what is obtained is often surprisingly accurate, and thus quite useful for engineering analyses. We first derive the control-volume continuity equation from the differential equation, and we then apply this to several examples. In such practical applications it is sometimes helpful to deal with finite volumes, often called control volumes , having bounding surfaces called control surfaces , as in basic thermodynamics. Control volumes and their associated control surfaces are selected (defined) for convenience in solving any given problem, and while there usually is not a unique way to define the control volume the ease with which any particular problem might be solved can depend very strongly on that choice. Furthermore, it is important to note that a control volume does not necessarily coincide with the “system” defined earlier, as will be apparent as we proceed. Derivation of Control-Volume Continuity Equation Since Eq. (3.19) must hold at every point in a fluid, if we now take R to be a control volume (rather than a fluid element as done earlier) we can write integraldisplay R ( t ) ∂ρ ∂t + ∇ · ρ U dV = 0 ,
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62 CHAPTER 3. THE EQUATIONS OF FLUID MOTION which is the same as Eq. (3.18), but our interpretation of R now has changed. We next use Gauss’s theorem “in reverse” to convert the second term back to a surface integral. This leads to integraldisplay R ( t ) ∂ρ ∂t dV + integraldisplay S ( t ) ρ U · n dA = 0 . Finally, we apply the transport theorem, Eq. (3.9) with F = F = ρ , to obtain d dt integraldisplay R ( t ) ρdV = integraldisplay S ( t ) ρ U · n dA + integraldisplay S ( t ) ρ W · n dA , or d dt integraldisplay R ( t ) ρdV + integraldisplay S ( t ) ρ ( U W ) · n dA = 0 . (3.22) It is worthwhile to note that U is the flow velocity in the control volume R ( t ), and W is the velocity of the control surface S ( t ). Furthermore, as already indicated by Eq. (3.15), the first term is just the time-rate of change of mass in the control volume R ( t ). In Eq. (3.22) it is useful to divide the control surface area into three distinct parts: i ) S e ( t ), the area of entrances and exits through which fluid may enter or leave the control volume; ii ) S m ( t ), the area of solid moving surfaces; and iii ) S f ( t ), the area of solid fixed surfaces. Observe that any, or all, of these can in principle change with time. Figure 3.5 provides an example showing these different contributions to the control surface S . In this figure we see that the valves make up the parts of S corresponding to entrances and exits and are labeled S e . The cylinder head and sidewalls are fixed and denoted by S f ; but it should be noted that the actual area corresponding to the surface of the control volume R ( t ) is changing with time; and the exposed area of the sidewalls comprise a major contribution to R ( t ). Finally, the moving piston surface contributes an area S m . In the present case this area is moving, but it is
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